What is the general solution for 2+cos(2x)=-5sin(x),0<= x<= 2pi ?

Q: What is the general solution for 2+cos(2x)=-5sin(x),0<= x<= 2pi ?

The general solution for 2+cos(2x)=-5sin(x),0<= x<= 2pi is x=(7pi)/6 ,x=(11pi)/6

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2+cos(2x)=-5sin(x),0<= x<= 2pi

Solution

2 + cos (2 x) = - 5 sin (x), 0 \leq x \leq 2 π

Solution

x = \frac{7 π}{6}, x = \frac{11 π}{6}

Degrees

x = 21 0^{\circ}, x = 33 0^{\circ}

Solution steps

2 + cos (2 x) = - 5 sin (x), 0 \leq x \leq 2 π

Subtract

- 5 sin (x)

from both sides

2 + cos (2 x) + 5 sin (x) = 0

Rewrite using trig identities

2 + cos (2 x) + 5 sin (x)

Use the Double Angle identity:

cos (2 x) = 1 - 2 sin^{2} (x)

= 2 + 1 - 2 sin^{2} (x) + 5 sin (x)

Simplify

= 5 sin (x) - 2 sin^{2} (x) + 3

3 - 2 sin^{2} (x) + 5 sin (x) = 0

Solve by substitution

3 - 2 sin^{2} (x) + 5 sin (x) = 0

Let:

sin (x) = u

3 - 2 u^{2} + 5 u = 0

3 - 2 u^{2} + 5 u = 0 : u = - \frac{1}{2}, u = 3

3 - 2 u^{2} + 5 u = 0

Write in the standard form

a x^{2} + b x + c = 0

- 2 u^{2} + 5 u + 3 = 0

Solve with the quadratic formula

- 2 u^{2} + 5 u + 3 = 0

Quadratic Equation Formula:

For

a = - 2, b = 5, c = 3

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 2 ) \cdot 3}{2 ( - 2 )}

5^{2} - 4 (- 2) \cdot 3 = 7

5^{2} - 4 (- 2) \cdot 3

Apply rule

- (- a) = a

= 5^{2} + 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- 5 \pm 7}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- 5 + 7}{2 ( - 2 )}, u_{2} = \frac{- 5 - 7}{2 ( - 2 )}

u = \frac{- 5 + 7}{2 ( - 2 )} : - \frac{1}{2}

\frac{- 5 + 7}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 5 + 7}{- 2 \cdot 2}

Add/Subtract the numbers:

- 5 + 7 = 2

= \frac{2}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{- 4}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

u = \frac{- 5 - 7}{2 ( - 2 )} : 3

\frac{- 5 - 7}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 5 - 7}{- 2 \cdot 2}

Subtract the numbers:

- 5 - 7 = - 12

= \frac{- 12}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 12}{- 4}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3

The solutions to the quadratic equation are:

u = - \frac{1}{2}, u = 3

Substitute back

u = sin (x)

sin (x) = - \frac{1}{2}, sin (x) = 3

sin (x) = - \frac{1}{2}, sin (x) = 3

sin (x) = - \frac{1}{2}, 0 \leq x \leq 2 π : x = \frac{7 π}{6}, x = \frac{11 π}{6}

sin (x) = - \frac{1}{2}, 0 \leq x \leq 2 π

General solutions for

sin (x) = - \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solutions for the range

0 \leq x \leq 2 π

x = \frac{7 π}{6}, x = \frac{11 π}{6}

sin (x) = 3, 0 \leq x \leq 2 π :

No Solution

sin (x) = 3, 0 \leq x \leq 2 π

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = \frac{7 π}{6}, x = \frac{11 π}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for 2+cos(2x)=-5sin(x),0<= x<= 2pi ?
The general solution for 2+cos(2x)=-5sin(x),0<= x<= 2pi is x=(7pi)/6 ,x=(11pi)/6